Integrable systems and differential Galois theory. (English) Zbl 1418.37096

Morales-Ruiz, Juan J.

Bolsinov, Alexey et al., Geometry and dynamics of integrable systems. Lecture notes for the advanced course, Barcelona, Spain, September 2013. Edited by Eva Miranda and Vladimir Matveev. Basel: Birkhäuser/Springer. Adv. Courses in Math., CRM Barcelona, 1-33 (2016).

Summary: At the end of the nineteenth century, E. Picard [C. R. Acad. Sci., Paris 96, 1131–1134 (1883; JFM 15.0258.02); Toulouse Ann. 1, A1–A15 (1887; JFM 19.0308.01); Traité d’analyse. T. III. 3 éd. Paris: Gauthier-Villars (1928; JFM 54.0450.09), Chapter XVII] and, in a clearer way, M. E. Vessiot in his PhD Thesis [Ann. Sci. Éc. Norm. Supér. (3) 9, 197–280 (1892; JFM 24.0283.01)], created and developed a Galois theory for linear differential equations. This field of study, henceforth called Picard-Vessiot theory, was continued from the forties to the sixties of the twentieth century by E. R. Kolchin, through the introduction of the modern algebraic abstract terminology and the obtention of new important results, see [Differential algebra and algebraic groups. New York-London: Academic Press (1973; Zbl 0264.12102)] and references therein. Today, the standard reference of this theory is the monograph [M. van der Put and M. F. Singer, Galois theory of linear differential equations. Berlin: Springer (2003; Zbl 1036.12008)].
For the entire collection see [Zbl 1357.37002].

MSC:

37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J30	Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
34M50	Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
12H05	Differential algebra
34M15	Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain

Keywords:

Hamiltonian system; Riccati equation; Galois group; Galois theory; identity component

Citations:

Zbl 0264.12102; Zbl 1036.12008; JFM 15.0258.02; JFM 19.0308.01; JFM 54.0450.09; JFM 24.0283.01